Coordinate transformation computer



Feb. 19, 1963 G. R. GRADO COORDTNATE TRANSFORMATION COMPUTER 7 Sheets-Sheet 1 Filed Sept. 25, 1959 Gilbert R. Grado,

IN V EN TOR.

Feb. 19, 1963 G. R. GRADO 3,078,042

COORDINATE TRANSFORMATION COMPUTER Filed Sept. 23, 1959 '7 Sheets-Sheet 2 NORMAL PROJECTION OF LONGITUDINAL BODY AXIS IN F., F (HORIZONTAL PLANE.)

FIG.5.

PLANE VERTICAL Gilbert R. Grado,

Feb. 19, 1963 G. R. GRADO COORDTNATE TRANSFORMATION COMPUTER 7 Sheets-Sheet 3 Filed Sept. 25, 1959 Silbe?? RGmdo,

INVEN TOR.

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Feb. 19, 1963 G. R. GRADO COORDINATO TRANSFORMATION COMPUTER Filed Sept. 25, 1959 7 Sheets-Sheet 4 @moo Silber? R. Grado,

Feb. 19, 1963 G. R. GRADO OOOROINATR TRANSFORMATION COMPUTER 7 Sheets-Shree?I 5 Filed Sept. 23, 1959 INVEN TOR. ag zia;

Feb. 19, 1963 7 Sheets-Sheet 6 Filed sept. 25, 1959 C; Amy

Feb. 19, 1963 G. R. GRADO 3,078,042

COORDINATE TRANSFORMATION COMPUTER Filed Sept. 23, 1959 7 Sheets-Sheet '7 cF (CR) Gilben` R. Grado,

JNVENToR.

BY J6'. 4'.

-engine mounts. (heading, elevation angle, or other angles depending upon angles.

`tions are repeated, e.g. yaw, pitch and roll.

3,@786d2 Patented Feb. 19, 1963 tice CRDINATE TRANSFGRMATON CGMPUTER Gibert R. Grado, Ei Paso, Tex., assigner tothe United States of America as represented by the Secretary of the Army Fitted Sept. 23, 1959, No. 841,910 7 Ciairns. (Cl. 23S-187) (Granted under Title 35, US. Code (1952), sec. 266) This invention described herein may be manufactured and usedby or for the Government for governmental purposes without the payment of any royalty thereon.

This invention relates to electronic computers and particularly to a computer for solving problems involving the transformation from one set or frame of spatial coordinate references to another.

It is an object of this invention to provide a computer which requires fewer components and less time to program.

The novel features that are considered characteristic of this invention are set forth with particularity in the appended claims. The invention itself, however, both as to its organization and method of operation as well as additional objects and advantages thereof will best be understood from the following description when read in connection with the accompanying drawing in which:

.FIGURES 1-4 illustrate the nature of the problems to be solved by the invention.

FIGURE 5 shows a block diagram of an embodiment of the invention.

FIGURES 6-9 show in detail elements of an embodiment of the invention.

As is well known the location and motion of an object in space, either real or hypothetical, may be conveniently plotted with a respect to three perpendicular axes rigidly fixed with respect to each other. A set of these axes, which we will refer to as a frame, may as a whole be oriented to best serve the problem at hand. Examples of these reference frames, particularly as applied to space vehicles and guided missile weapons, are shown in FIG- URE l.

The angular relationship between various three dimensional reference frames associated With missile weapon systems is one of the basic required studies necessary in system synthesis and system analysis. For, in the design of a weapon system wherein a guided missile is to be automatically vectored to a target, innumerable vector resolutions are required. For example, a Doppler-inertial guidance system requires transformations of Doppler velocity along carrier axes to components of Doppler velocity along platform axes. Angular relations are necessary for the alignment of slaved platforms to a master platform for air-launched ballistic missiles, and for the stabilization of radar, cameras, star-tracker, or gimballed Knowledge of the missile attitude the type of flightpath control used) can be obtained from gyros and stable platforms. ln system simulation studies,

-various types of coordinate transformations are necessary, some of which in actual ight are achieved physically.

One reference frame may be uniquely oriented with respect to a second reference frame through three ordered These are termed Euler angles and are of two types:

(l) Repetitive Euler angles as used in classical mechanics in which one of the three angular rotations is repeated and a line of nodes is established, e.g. the three ordered rotations might be yaw, roll, yaw.

(2) Successive Euler Angles in which none of the rota- The first rotation is about an axis of the one (initial) reference frame, the third of the ordered rotations is about an axis of the other (final) reference frame, and the second rotation is about an axis normal to the rst and third rotation axes.

The roll axis as contemplated herein is the 151 or 'Int-1; (roll axis after jth rotation vector). For example if the first rotation is a roll, then the roll axis -is F1, if the first rotation is either pitch or yaw and the second rotation roll, then 11 is the roll'axis, that is, the roll axis after the first rotation. lf the last rotation is roll then the roll is R`12=`1 (final position of roll axis). Similar considerations hold for'the pitch axis and yaw axis.

Thus it is seen that the angles r (roll), p (pitch), y (yaw) respectively are measured in variable planes 'perpendicular to their axes of rotation, where the axis of rotation is a function of the Euler sequence used.

Gimballed body (c g. gyros, stable platforms, and radar antennas) pick off angles are of the successive type and consequently extensively used in missile equations of motion. Therefore, the Euler angles referred to herein are to be understood to be successive type Euler angles.

Consider the orientation of a unit vector 'E1 in the 'F1 reference frame (FIGURE 2). As shown, the three angles that E1 Vmakes with the F1, "2 and '153 axes respectively are given by a1, IE21 and 'y1. The direction cosine between the E1 and F1 unit vector is cos a1=11 In terms of Euler angles (for example radar coordinatesazimuth angle y and elevation angle p) the relationship between ce1, y and p is cos a1=cos y cos p.

The total number of ordered sets or sequences of Euler angles is given by the permutations of three angles taken three at a time, or six. The angles used are r, p, y, Where the correspondence vis such thatr, p, y are always associated with rotations about the 15, 25, 31 (j=1, 2, 3) vectors, respectively. The 'sequences 'of rotations are:

The Euler Angles represent an ordered triple of numbers, thus for a unique orientation of one frame (E1 with respect to a second frame TF1) the angles y, p and r for the sequence (y1, p1, r1) will have different numerical values than the angles y2, r2 and p2 for the sequence (y2, r2, p2). This is shown in FIGURE 3 where the yaw angle and elevation angles for thev first sequence represent respectively the normal projection of the E1 axis in the F1, 'F2 plane (heading angle for Zero angle of attack) and the pitch angle is the elevation plane pitch angle. VFor the second sequence (y2, r2, p2), y2 is not the heading angle and p2 is a plane of symmetry pitch angle if a symmetrical air frame were assumed with E1 longitudinal axis, and R2 a left wing axis. Thus, unless an Euler sequence is specified, the three commonly denoted angles yaw, pitch, and roll are `without specific meaning.

By determining the direction cosine matrixvfor each of the angular rotations in turn for a particular sequence, and vmultiplying these matrices together, the direction cosine matrix for the overall transformation of vectors from one frame to another can be obtained. Forexample, the matrices for the sequences y, p, r and y, r, p

are as follows:

By interchanging the rows and columns of the matrix Vand changing the signs of the angles we may solve for F1, F2 and F3 and thus perform the inverse vector transformation. A

Rewritten in this form we have:

Sequence r, y, p

As will be appreciated from the foregoing there are a total of 54 matrix elements (27 repeated with opposite sign) involved in the 6 sequences of rotation.

In the simulation of the guided missile system it also becomes necessary to generate the Euler angles. The information available consists of the initial angle plus the relative angular velocities of the rotating frame with respect to the xed reference frame. The missile studies up to the present have used the xed reference frame as an inertial frame, and the equations relating the motions between the two frames are shown as follows for the following two of the six sequences.

Sequence y, p, r

Sequence y, r, p

Where y, 1i, 1'* represent the angular velocity of y, p, r and Pr, Qr and Rr are the three angular velocities about El, E2 and R3, respectively.

Upon examination ofthe direction cosine matrices for the various sequences, a definite pattern has been discovered. Even though the angles are referred to by different names and the axes numbered in a different manner, there is always a rotation about an axes of the first reference frame, a second rotation about an axis normal to the first and third rotation axes, and a third rotation about an axis of the final reference frame.

With reference to an analysis of the ramifications of this finding a unique computing system has been devised. It fulfills the object of providing a means of vector transformation comprising fewer components and requiring much less programming time.

In accordance with the invention means are provided to generate three Euler angles and sine and cosines thereof, the iirst angle da from the rst rotation which is about an axis of a first reference frame, the second angle sib from the second rotation about an axis normal to the first and third axes, and the third angle )Ic from the third rotation which is about an axis of a final reference frame. Next, the sine and cosine components are uniquely corn` bined to obtain as a function of )a, )I b and )c, a matrix, the elements of which when multiplied by corresponding vectors along axes of a first frame produces vectors along axes of a second frame of axes. Finally means are provided to perform this multiplication and to appropriately combine all vector components from the first frame to form the three resulting ortransformed vectors in the second frame.

Referring now to FIGURE 5 illustrating the complete computer there is shown Euler angle and sine-cosine gen- Verator 1t) to which is applied initial Euler angles ao,

Cbo and dico and rates of rotation w1, m2 and w3 which represent the angular rotation about the first axis of rotation, the second axis of rotation and the third axis of rotation, respectively, of the rotating frame (FIGURE 4). As illustrated in FIGURE 6, generator 10 solves these equations for Euler angle rates d, b and e':

In order to program the computer in terms of the actual Euler angles yaw, pitch and roll it is only necessary to Aconsider them da, b and dic in accordance with the sequence in which they occur. As illustrated in FIGURE 4 for the sequence yaw (y), pitch (p) and roll (r). The yaw angle is designated a, the pitch angle b and roll angle sie.

Next the Euler angles rates are integrated with respect to time in integrators 16, 18 and 26 and combined with the initial Euler angles to provide the instantaneous Euler angle outputs )1a, sib and dic.

Finally generator 1f) responsive to voltages proportional to )a, lb and sie generates function signals which are the sines and cosines of these angles. These functions are applied as an input to vector conversion factor generator 12, shown in detail in FIGURE 8. A matrix of nine vector conversion factors are generated and these factors are sufficient to transform vectors from one frame to another regardless of the sequence of rotation.

The output of generator 12 is applied to vector transformation computer 14, which is shown in detail in FIG- URE 9. This computer is designed to multiply each vector in a first frame by the three conversion factors which produce the components of this vector along each of the three axes of the second frame. Second, the computer adds all like component vectors so produced to obtain the three final vectors in the second frame. FIG URE 2 and the discussion thereof illustrates the derivation of the sine-cosine factor necessary to transform one vector of a first frame to a component of a final frame.

Considering now the Euler angle and sine-cosine generator shown in FIGURE 6, it is to be noted that the identically labeled diamond shape terminals are interconnected. The round terminals indicate a generator input. The value s is solved by adding (or subtracting depending upon Ithe sign of the quantity) in amplifier 22 the product wz sin c obtained in multiplier 24 and the product w1 cos c obtained in multiplier 26. The sum is then divided in Vdivider 2.8 by cos b (in effect multiplied by sec b). The

value (i is then applied through the upper (or lower) terminal of switch 30 to integrator 16 which produces da. Angle a from integrator 16 is supplied to sinecosine generator 32 which produces sines yand cosines of )C a. Flip-Hop 34 is also responsive to da, the output of integrator 16, is triggered when da approaches dr or -rr and causes switches 35i and 36 to operate causing a reversal in phase of the voltages these switches supply due to the presence of unity gain phase reversing amplifiers 38 and 4t). SwitchY 42 is provided for the instance in which the Euler angles are generated externally and a is known. In this case the moving contact is moved to the lower position and )a0 is directly supplied to sinecosine generator 32. Gtherwise 9210 is added to the generated component of ga in integrator 16.

The value b is solved by adding the product m2 cos c obtained by multiplier 44 'through amplifier 46 in amplifier 48 to the product w1 sin c obtained by amplifier 49. The value b is fed to integrator 18 Ito obtain b which in turn is supplied sine-cosine generator 5t) which gencrates the sines and cosines of sib. Switch 52 provides -for an initial Euler angle {bo to be added to the angle generated or supplied directly to sine-cosine generator 50. VThe operation of the sine-cosine generator for )Ib1 to 30 and within .02% up to 90.

the second angle in the Euler sequence, is limited to less than i90 degrees and thus there is no provision for reversing the phase of angle ,b or the sine of angle )1b. This is compatable with most operations since 90 physically symbolizes gimbal lock in a gyro. The value c' is obtained by adding w3 in amplifier 54 to the product of the output of divider 28 and sin b obtained from multiplier 56. The value c' is fed to integrator 20 directly or through unity gain phase reversing arnpliiier 58. The output of integrator 20, c+ Ico or Ico alone by virtue of switch 67 is fed to sine-cosine generator 60 which produces sines and cosines of lc. Flip-flop 62 serves to activate switches 64 and 66 to reverse the sign to )c and sine c by virtue of unity gain phase reversing amplifiers 58 and 63 when the voltage representative of capproaches im The incorporation of this feature with respect to the first da), and third (dic) angles allows continuous rotation of these angles to be observed. Switches 42, 52, and 67 are actuated simultaneously.

To examine in detail the sine-cosine generator em- -ployed as sine-cosine generators 32, Sil and 62 reference is made to FIGURE 7. The generator is supplied an input as illustrated in which zero degrees is rep-resented by zero voltage, 1r is represented by a maximum positive voltage and 1r by a maximum negative voltage. This input is fed through phase reversing limited ampliiier Astage 70 comprising amplier 78 having a gain of 2 and .limiter S0. The output is combined through load resistor 72 with the original input through load resistor 74. Due to the limiting effect at f11-/2 and -11-/2 the results is a sawtooth wave voltage. This voltage is fed to shaping -ampliiier stage 76. This stage lis provided with amplifier 82 and a series of diode (8M-resistor (86) negative Vfeedback paths topped otf between equal value resistors 88 forming a load circuit. The result is that as the output voltage increases in magnitude either positive or negative the amplifier gain decreases in steps or breakpoints to provide an output which approximates a sine wave. The

breakpoints are set at equal intervals and when set at six degree intervals an accuracy Vof .01% is provided up Thus the maximum accuracy is provided Where the sine values are smallest. This is important where a number of multiplications are to be performed. Assume for example that instead of a relatively constant percent error as is the case here, tolerances are, as is common in terms of a percent of full scale, which of course represents a decreasingvpercent of error, maximum at low values and minimum at large values. Assume further that an error of .01 of full scale is provided and that the range of response is between 0 and 100. At 100 the error of 1 may be insignificant but at an indicatedpvalue of 2, the error could be 50%.

If the indicated value were to be multiplied by, say a value of 80, the product could be from 80 to 160, an intolerable range of error. By providing a generator in which as the quantity decreases the error decreases, this lsituation is avoided; and as in .the case here, extreme accuracy over several multiplications may be obtained. The error of the generator just described is approximately equal to the sine of the angle.

The cosine generator employed an identical amplifier '90 and identical positive and negative voltage feedback networks Z1 and Z2 to those shown for the sine generator.

Their operation is the same, except of course the voltage `must be shifted 90, which i-s accomplished by an abso- ,lute value input circuit and an oppositely poled reference voltage. The absolute value of the angular input voltage is provided by applying the input voltage to two paths,

The outputs of the diodes are connected 6 at which point it is combined with an oppositely poled reference voltage applied through resistor 106. The resultant sawtooth voltage is shown.

The Vector Conversion Factor Generator `shown in FIGURE 8 and Vector Transformation Computer shown in FIGURE 9 comprise computing elements arranged to satisfy the following equations between vectors of a first reference frame and vectors of a second reference frame and thus provide a mechanism for vector transformation from the one frame to the other regardless of the sequence of angles defining the two frames.

B 5 R sin c sin b sin a-l-cos c cos a 2 noa cos b sin a) 7 F R eos c sin b cos a-l-sm c sin a sm c sin b cos a-I- cos c sin a 1 CR cos c cos a) The numbers over each sinedcosine function correspond Vto the designations set forth in FIGURE 9. Employing them rather than the 'functions themselves, the inverse transformation equations may be Written as follows:

As illustrated in FIGURE 4 AF, BF and CF are vectors associated, respectively, with first, second and third axes of rotation of a irst coordinate framey and AR, BR and CR are transformed vectors associated with the same axes after the axes have been rotated by a first rotation ):z, a second rotation b and a third rotation c.

Referring speciiically to FIGURE 8, the first term is obtained by multiplying cos a and cos b in multiplier 1110. The second term is obtained by multiplying sin a by cos b in multiplier i12. The third term is obtained by adding in amplifier 114i the product sin c cos a sin b obtained from multipliers 1116 and `1213 and product sin va cos c obtained from multiplier 120. The fourth term is obtained by adding in amplifier 122 the product sin a sin b sin c in multipliers 124, 126 with the product cos va cos c obtained through ampliiier 130 from multiplier 123. rlhc fifth term is simply-sin b. The sixth term is the sum of the output of multiplier 124 obtained in amplifier E32 through amplifier 134 and the product sin b cos a cos c obtained vfrom multipliers 136 and 123. The seventh term is obtained by adding in amplifier 138 the output of multiplier 116 and the product sin a cos c sin b from multipliers and 140. The eighth term is obtained from multiplier 142 as the product cos b sin c. The ninth term is the product cos b cos c obtained from multiplier 144. inasmuch as the computations illustrated are for negative angles to compute from lhe rotated frame back to the original frame, the sign of all the terms is negative. The above mentioned multipliers may be an Electronic Associates Time Division Electronic Multiplier Model #7.006. Y

Referring to FIGURE 9 the computer circuit is labeled for conversion from an original frame to a generated frame and from a generated frame back to an original frame. The terminal interconnections to FIGURE 8 and other inputs for conversion from an initial frame to a generated frame are in parenthesis. In accordance with the equations set forth above, multipliers 150, 152

and 154 multiply terms 1-3 by CR, (or terms 1, 4 and 7 by CF), multipiiers 156, 158 and 160 multiply terms 4-6 by BR (or terms 2, and 8 by BF), and multipliers 162, 164 and 166 multiply terms 7-9 by AR (or terms 3, 6 and 9 by AF). As modified, terms 1, 4 and 7 (or terms 1, 2, 3) are then added in amplifier 16S to obtain CF (or CR), terms 2, 5 and 8 (or terms 4, 5 and 6) are added in amplier 176 to obtain BF (or BR), and terms 3, 6 and 9 (or terms 7, 8 and 9) are added in amplifier 172 to obtain AF (or AR). In the parenthetical case the Vangles will be positive and the negative sign Vdropped from eaoh term.

While the foregoing is a description of the preferred embodiment, the following claims are intended to include those modications and variations that are within the spirit Yand `scope of my invention.

The following invention is claimed:

1. A three dimensional coordinate transformationy computer comprising first, second and third function generating means each responsive to a voltage proportional to an angle for generating sines and cosines o-f that angle, a first angle source means for supplying to said rst function generating means a voltage which is proportional to a first angle )a generated by rotation ofthe first axis of a first coordinate lreference frame, a'second angle source means for supp-lying to said second function generating means a voltage which is proportional to a second angle ib generated by rotation of a second axis normal to both said first axis and a third rotation axis, a third angle source means for supplying to said third function generating means a voltage Which is proportional to a third angle )c generated by rotation about said third rotation axis which third axis is an axis of a `second coordinate reference frame, first computing means responsive to the output of said first, second and third function generating means for computing at least the terms:

(1) cos a cos b (2) sin a cos b (3) sin b (4) sin c cos a sin b-sin a cos c (5) sin c sin a sin b+cos a cos c (6) cos b sin c (7) sin c sin a+sin b cos a cos c (8) sin c cos a-l-sin a cos c sin b (9) cos b cos c said terms being grouped in a plurality of predetermined sets each comprising three of said terms, second computing means responsive to the output of said first corn-V puting means comprising first product means for selectively multiplying one of a first pair of said predetermined sets by a first quantity to produce first, second and third product outputs, second product means for selectively multiplying one of a second pair of said predetermined sets by a second quantity to produce first, second and third product outputs, third product means for selectively multiplying one of a third pair of said predetermined sets by a third quantity to produce first, second and third product outputs, first summing means for adding said first product outputs of said product means, second summing means for adding said second product outputsY of said product means, andY third summing means for adding said third product outputs of said product means.

2. The computer set forth in claim 1 wherein said first computing means is connected to said second computing means to provide to said first product means as said first set of three terms, terms 1, 2, 3 to provide to said second Vproduct means as said second set of three terms, terms quantity is a vector associated with the second angle axis of said second coordinate frame, and said rst quantity is a Vector associated with the third angle axis of said second coordinate frame.

3. The computer set forth in claim 1 wherein said first computing means is connected to said second computing means to provide to said first product means as said first set of three terms, terms l, 4, 7, to provide to said second product meansas said second terms, terms 2, 5, 8, and to provide to said third product means as said third set of three terms, terms 3, 6, 9, said product means inputs and outputs being related by the order in which they appear, and said third quantity is a vector associated with the first angle axis of said first coordinate frame, said second quantity is a vector associated with the second angle axis of said first coordinate frame, and said first quantity is a vector associated with the third angle axis of said first coordinate frame.

4. The computer set forth in claim `=1 wherein said function generators each comprise a sine generator and a cosine generator, said angle source means provides a voltage output having a maximum value of one polarity corresponding to ir and an equal maximum value of an opposite polarity corresponding to -1r, said sine generator comprising a first input means responsive to said angle source means for generating a sawtooth waveform with opposite peaks of said waveform at i1r/2, said cosine generator comprising a second input means responsive to said angle source means for generating a sawtooth waveform wherein the peaks of said waveform occur of one polarity at ii" and of the other polarity at :1 -1r, a first wave shaping means responsive to said first input means and a second Wave shaping means responsive to said second input means, each of said wave shaping means comprising a feedback amplifier having a plurality of voltage responsive inverse feedback loops providing increased feedbacks in equal steps as the output voltage magnitude, positive or negative, increases.

5. The computer set forth in claim 4 wherein said feedback loops comprise first and second voltage divider networks each having a plurality of equal resistance impedance elements, said first divider network being connected between the output of said feedback amplifier and a source of positive voltage, said second divider network being connected between the output of said feed'- back amplifier and a source of negative voltage, the input of said feedback amplifier being connected through rectifier-resistance paths to interconnections of said impedance elements, said rectifiers connected to said first divider network being poled to permit current fiow toward said first `divider network and those connected to said second divider network being poled to permit fiow away from said second divider network.

6. The computer set forth in claim 5 wherein said first angle source means comprises a first multiplier for providing w1 cos c, a second multiplier for providing wz sin c, first adding means responsive to said first and second multiplirs for providing w1 cos c-l-w2 sin c, dividing means responsive to the output of said first summing means for dividing said last named output by cos b, first integrating means responsive to the output of said dividing means for providing an output a to said rst function generating means, first phase reversing means including a switch and a unity gain phase reversing amplifier connected between said dividing means and said first integrating means for providing either phase of the output of said dividing vmeans, second phase reversing means including a switch and a unity gain phase reversing amplifier connected to the sine output of said first Vfunction generating means for providing either phase of the sine da, switching means responsive to the output of said integrating means and connected to said switches of said first and secondV phase reversing means for causing phase reversal whenever a approaches if, said second angle source means comprises a third multiplier for providing el sin c, a fourth 9 multiplier for providing u2 cos c, a second adding means responsive to the output of said third and fourth multipliers for providing wz cos c-l-wl sin c, second integrating means responsive to the output of said second adding means for providing an input b to said second function generating means; said third angle source means comprises a fth multiplier responsive to sin b and the output of said rst divider, a third adding means responsive to w3 and the output of said fth multiplier, third integrating means responsive to the output of said third adding means for providing an input dic to said third function generating means, third phase reversing means including a switch and a unity gain phase reversing amplifier connected between said third adding means and said integrating means or providing either phase of the output of said third adding means, fourth phase reversing means comprising a switch and a unity lgain phase reversing amplifier connected to the sine output of said third function generating means for providing either phase of sin c, switching means responsive to the output of said third integrating means and connected to the switches of said third and fourth phase reversing means for producing phase reversal whenever c approaches i-fr and wherein said w1, wz, w3 are as defined in the specification.

7. The computer set forth in claim 6 wherein said angle sources each comprise means for selectively adding an initial angle to the output of said integrating means or supplying an externally generated angle to said function `generating means.

Magnin Sept. 29, 1959 Slater July 19, 1960 

1. A THREE DIMENSIONAL COORDINATE TRANSFORMATION COMPUTER COMPRISING FIRST, SECOND AND THIRD FUNCTION GENERATING MEANS EACH RESPONSIVE TO A VOLTAGE PROPORTIONAL TO AN ANGLE FOR GENERATING SINES AND COSINES OF THAT ANGLE, A FIRST ANGLE SOURCE MEANS FOR SUPPLYING TO SAID FIRST FUNCTION GENERATING MEANS A VOLTAGE WHICH IS PROPORTIONAL TO A FIRST ANGLE $A GENERATED BY ROTATION OF THE FIRST AXIS OF A FIRST COORDINATE REFERENCE FRAME, A SECOND ANGLE SOURCE MEANS FOR SUPPLYING TO SAID SECOND FUNCTION GENERATING MEANS A VOLTAGE WHICH IS PROPORTIONAL TO A SECOND ANGLE $B GENERATED BY ROTATION OF A SECOND AXIS NORMAL TO BOTH SAID FIRST AXIS AND A THIRD ROTATION AXIS, A THIRD ANGLE SOURCE MEANS FOR SUPPLYING TO SAID THIRD FUNCTION GENERATING MEANS A VOLTAGE WHICH IS PROPORTIONAL TO A THIRD ANGLE $C GENERATED BY ROTATION ABOUT SAID THIRD ROTATION AXIS WHICH THIRD AXIS IS AN AXIS OF A SECOND COORDINATE REFERENCE FRAME, FIRST COMPUTING MEANS RESPONSIVE TO THE OUTPUT OF SAID FIRST, SECOND AND THIRD FUNCTION GENERATING MEANS FOR COMPUTING AT LEAST THE TERMS: (1) COS A COS B (2) SIN A COS B (3) -SIN B (4) SIN C COS A SIN B-SIN A COS C (5) SIN C SIN A SIN B+COS A COS C (6) COS B SIN C (7) SIN C SIN A+SIN B COS A COS C (8) -SIN C COS A+SIN A COS C SIN B (9) COS B COS C SAID TERMS BEING GROUPED IN A PLURALITY OF PREDETERMINED SETS EACH COMPRISING THREE OF SAID TERMS, SECOND COMPUTING MEANS RESPONSIVE TO THE OUTPUT OF SAID FIRST COMPUTING MEANS COMPRISING FIRST PRODUCT MEANS FOR SELECTIVELY MULTIPLYING ONE OF A FIRST PAIR OF SAID PREDETERMINED SETS BY A FIRST QUANTITY TO PRODUCE FIRST, SECOND AND THIRD PRODUCT OUTPUTS, SECOND PRODUCT MEANS FOR SELECTIVELY MULTIPLYING ONE OF A SECOND PAIR OF SAID PREDETERMINED SETS BY A SECOND QUANTITY TO PRODUCE FIRST, SECOND AND THIRD PRODUCT OUTPUTS, THIRD PRODUCT MEANS FOR SELECTIVELY MULTIPLYING ONE OF A THIRD PAIR OF SAID PREDETERMINED SETS BY A THIRD QUANTITY TO PRODUCE FIRST, SECOND AND THIRD PRODUCT OUTPUTS, FIRST SUMMING MEANS FOR ADDING SAID FIRST PRODUCT OUTPUTS OF SAID PRODUCT MEANS, SECOND SUMMING MEANS FOR ADDING SAID SECOND PRODUCT OUTPUTS OF SAID PRODUCT MEANS, AND THIRD SUMMING MEANS FOR ADDING SAID THIRD PRODUCT OUTPUTS OF SAID PRODUCT MEANS. 